3 Answers
3

Thinking in depth about it, either could be considered correct (i.e. neither is positively incorrect), and both are common. It's true that an article should often be used before a noun. Hence the following is incorrect:

I accidentally jogged the arm of person walking by.

It should be:

I accidentally jogged the arm of a person walking by.

Or, if you have already been introduced to that person:

I accidentally jogged the arm of the person walking by.

However, since phrases like what kind are referring to a type, not an individual, the article a is not mandatory. Consider the phrase what kind of dog. Dog is a type here, not a specific dog.

On what basis do you say what kind of person is "the correct" form, with the implication there's something incorrect/less correct about including an article? I know the latter has always been less common (not so markedly a century or two ago), but I can't see there's any principle of grammar involved there. It goes the other way with some nouns anyway (compare What kind of climate with What kind of a climate), and I doubt you could set a "cut-off" point to say which nouns "need" an article in this construction.
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FumbleFingersMar 16 '12 at 17:24

I've just edited so as to remove my tentative hypothesis, but unfortunately I can't come up with a convincing reasoning why what kind of a is definitely grammatical.
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DanielMar 16 '12 at 18:27

@Daniel: You have my upvote simply for the revised answer (since I can no longer find anything to carp at!). I just came up with "climate" off the top of my head, but I'd have to say I agree with the majority of writers in Google Books that an article is "better" there. Perhaps some other (countable) noun would illustrate that preference even more markedly, and might guide us towards some underlying principle.
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FumbleFingersMar 16 '12 at 18:38

The conclusion that a isn't mandatory for the type makes sense, but you've failed to account for why it's not plural.
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Brett ReynoldsMar 17 '12 at 19:05

The question form complicates this problem. Let's start by looking at this kind of X and these kinds of X. There seems to be some sort of concord between the number of the determiner (i.e., this/these) and the number of X. That is, you would expect this kind of cow or these kinds of cows but this kind of cows is rare and maybe even ungrammatical for some people, even though logically, cows would make sense since it is commonly used when referring to the general group (e.g., cows give milk). Consider also cows of this kind not *cow of this kind.

This concord suggests that the determiner for kind is bleeding through and filling the determiner slot for X too, which is why the noun in X can be singular without needing a determiner as other singular countable nouns typically do.

Going back to the original question, I think when people say what kind of cow, they take what to fill the obligatory determiner slot for cow and the kind of becomes somewhat transparent. At some level, they are asking what cow. This would be somewhat analogous to the subject-verb agreement in cases like what kind of cows were used (cf. what kind of cows was used).

Those who focus on kind, may be more prone to see the determiner slot for X as unfilled and fill it with a, just as they may force number agreement between kind and was in the example above.